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Ill-Posedness in Limited Discrete Fourier Inversion and Regularization for Quasi Distributions in LaMET

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Pith's one-line read This paper claims that the limited inverse discrete Fourier transform in the LaMET framework is a moderately ill-posed inverse problem—solutions exist and are unique, but stability fails—and that Tikhonov regularization with L-curve…

desk verdict Useful, workmanlike application of Tikhonov regularization to LaMET quasi-DA inversion, but the 'moderately tractable' spectral classification is asserted rather than shown. read the letter →

arxiv 2506.16689 v1 pith:XIH7I6JY submitted 2025-06-20 hep-lat

Ao-Sheng Xiong , Jun Hua , Ting Wei , Fu-Sheng Yu , Qi-An Zhang , Yong Zheng This is my paper · ORCID
classification hep-lat MSC 65F2265R30 PACS 12.38.Gc
keywords inverseproblemsTikhonovregularizationL-curvemethodlimiteddiscreteFouriertransformlarge-momentumeffectivetheoryquasidistributionamplitudelatticeQCDill-posedness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether the limited, noisy coordinate-space data available from lattice QCD can be inverted to recover a parton distribution inside the large-momentum effective theory (LaMET). It argues that the inversion is ill-posed in the strict Hadamard sense—existence and uniqueness hold, but tiny input noise is amplified by a factor of about $10^{17}$—yet that this ill-posedness is mild enough to be tamed by Tikhonov regularization. The paper demonstrates the method on synthetic toy models with both uncorrelated and correlated noise, and then on real lattice QCD data for the pion quasi distribution amplitude, where the regularized result agrees with the physics-driven $\lambda$-extrapolation method. If the claims hold, LaMET practitioners can invert limited Fourier data with first-principles uncertainty quantification, without fitting ansatz functions.

What carries the argument

The load-bearing object is the discrete Fourier matrix $K$ with entries $\exp(i x \lambda)$ on a finite grid, whose singular value decomposition $K = U \Sigma V^T$ exposes the instability: the formal solution $f = \sum_i (u_i^T g / \sigma_i) v_i$ divides by singular values as small as $10^{-18}$. Tikhonov regularization replaces that division with the normal equation $(K^\dagger K + \alpha I) f^\delta_\alpha = K^\dagger g^\delta$, adding $\alpha > 0$ to each singular value so that the noise-amplifying factors $\sigma_i^{-1}$ are controlled; the L-curve method fixes $\alpha$ by maximizing the curvature of the $\log\|K f - g^\delta\|^2$ versus $\log\|f\|^2$ tradeoff. Uniqueness of the limited Fourier transform is proved in the appendix by analytic continuation together with the Weierstrass approximation theorem.

What would settle it

Compute the singular values $\sigma_i$ of the real DFT matrix $K_{\mathrm{re}}$ for the paper's parameters ($x \in [-2,3]$, $\delta x = 0.01$; $\lambda \in [-20,20]$, $\delta \lambda = 0.5$) and plot $\log \sigma_i$ against the index $i$. A straight-line (linear) decay in this log-linear plot means geometric, i.e., exponential-in-index, decay—consistent with $\sigma_1/\sigma_{81} \approx 10^{17}$ giving a per-index ratio near $(10^{17})^{-1/80}\approx 0.61$—which directly contradicts the claim of slower-than-exponential decay. The classification of the problem as moderate rather than severe would then need to be revised.

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Extended reading notes

Core claim

The central claim is that the limited inverse discrete Fourier transform underlying LaMET is an ill-posed inverse problem of moderate severity. It satisfies the first two Hadamard criteria: the Paley-Wiener theorem gives existence for compactly supported parton distributions, and an appendix proves uniqueness on $L^2$ of a finite interval. Stability fails because the singular values of the DFT matrix decay to about $10^{-18}$, giving a condition number of order $10^{17}$ that amplifies input errors massively. Tikhonov regularization, with the regularization parameter chosen by the L-curve criterion, is shown to turn this unstable inversion into a stable optimization problem; regularized solutions reproduce the true profile in toy models and produce a pion quasi distribution amplitude from real lattice data consistent with $\lambda$-extrapolation. The paper additionally claims the singular spectrum decays more slowly than exponentially, placing this problem in a moderately tractable class distinct from severely ill-posed problems such as spectral-function reconstruction from Euclidean correlators.

Load-bearing premise

The classification of the LaMET inversion as 'moderately tractable' rests on the unproven assertion in Sec. II.B that the DFT matrix singular values decay more slowly than exponentially; the paper's own quoted spectrum ($\sigma_1/\sigma_{81} \approx 10^{17}$) is consistent with exponential decay, so if the decay is actually exponential or faster, the problem would belong to the more severe class and the claimed distinction would collapse.

Editorial extensions

If this is right

  • The LaMET Fourier inversion satisfies existence and uniqueness, so the only obstruction is stability, and Tikhonov regularization restores well-posedness.
  • L-curve-selected Tikhonov regularization provides quantified uncertainties for the pion quasi distribution amplitude without any ansatz-based functional form.
  • The regularized reconstruction from real lattice data is statistically consistent with the $\lambda$-extrapolation result, cross-validating the two independent methods.
  • Classifying the problem as moderately tractable separates it from severely ill-posed lattice QCD inversions (such as spectral-function reconstruction), indicating that standard regularization tools suffice here.
  • As the input error approaches zero, the regularized solution converges to the true distribution, giving a formal guarantee behind the numerical reconstructions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The 'slower than exponential' singular-value decay that underpins the moderate-tractability classification is not demonstrated quantitatively; the quoted ratio $\sigma_1/\sigma_{81} \approx 10^{17}$ is compatible with geometric decay in the index, which would put the problem in the severe class even though Tikhonov regularization might still work.
  • The same L-curve-regularized inversion could be applied as a cross-check to other LaMET observables (PDFs, GPDs, TMDs) that also provide only limited, noisy $g(\lambda)$ data.
  • The agreement between Tikhonov and $\lambda$-extrapolation could be turned into a quantitative diagnostic: plotting the difference between the two reconstructions as a function of $\lambda_{\max}$ and data precision would reveal where regularization bias or extrapolation systematic errors dominate.
  • Because the penalty term is $\|f\|^2_{\ell^2}$, the method favours smooth solutions and may smooth away genuine endpoint or kink structures; combining it with operator-product-expansion endpoint constraints is a natural next step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper analyzes the limited discrete Fourier inversion problem that arises in the large-momentum effective theory (LaMET) when reconstructing momentum-space partonic distributions from finite-range, discrete coordinate-space lattice data. It shows that the continuous inverse Fourier transform on a bounded support is unique (with a proof in Appendix A), but that the discretized inversion is numerically unstable, as quantified by a singular value condition number of order 10^17. The authors propose Tikhonov regularization with the L-curve method for selecting the regularization parameter, demonstrate its effectiveness on toy models with uncorrelated and correlated noise, and apply it to lattice QCD data for the pion quasi distribution amplitude from the Lattice Parton Collaboration. The reconstructed quasi-DA is reported to be consistent with the previously used lambda-extrapolation method. The paper claims that the limited Fourier inversion in LaMET falls into a class of 'moderately tractable' ill-posed problems, distinguished by slow (slower than exponential) singular value decay.

Significance. If the claims are correct, the paper would provide a mathematically principled alternative to physics-driven extrapolation for partonic distributions in LaMET, with the potential to quantify uncertainties without model assumptions. The paper contains a self-contained uniqueness proof, a clean toy-model validation with a known ground truth, and a real-data application. However, the central classification of the problem as 'moderately tractable' is not substantiated: the reported singular value ratio is fully consistent with exponential (geometric) decay, which is typically associated with severely ill-posed problems. The uncertainty quantification is also less first-principles than claimed, since the L-curve is a heuristic and the bootstrap procedure does not propagate the regularization parameter choice. The core demonstration, that Tikhonov regularization stabilizes the inversion, is plausible and valuable, but the overreaching framing and the absence of a spectral analysis prevent acceptance in the current form.

major comments (4)
  1. [Sec. II.B (after Eq. (14))] The statement that the singular values of K_re 'decay more slowly than exponentially' is load-bearing for the paper's classification of the problem as 'moderately tractable,' but it is not substantiated. The paper gives sigma_1 = 0.3545 and sigma_n = 3.8e-18 for n = 81, yielding sigma_n/sigma_1 ~ 1e-17. Over the 80 index steps, this ratio is perfectly consistent with geometric decay sigma_i ~ r^i with r ~ 0.61, i.e., exponential decay in the index. Algebraic (slower-than-exponential) decay would give a far milder condition number. Please provide a quantitative spectral analysis (e.g., the full singular value spectrum on a log scale, or an asymptotic bound) or remove the classification claim from the abstract and Sec. II.B.
  2. [Sec. II.B and Appendix A] The uniqueness theorem is proved for data given on a continuum interval [lambda_min, lambda_max], whereas the practical inversion uses finitely many discrete values of lambda. For finite discrete data, the inverse problem is generally underdetermined and uniqueness does not hold as stated. The manuscript should explicitly state that the existence/uniqueness results apply to the continuous idealization, and should discuss the relation between this idealization and the finite-dimensional discrete problem actually solved by the regularization.
  3. [Sec. IV, Eq. (27) and Fig. 9] The claim of 'first-principles uncertainty quantification' is not supported by the described methodology. The regularization parameter alpha is selected per bootstrap sample via the L-curve criterion, and the reported alpha values span three orders of magnitude (10^-6 to 10^-4). The final uncertainty band shown in Fig. 12 does not appear to include the systematic component from the ambiguity in alpha selection. Please describe how the uncertainty band is constructed, and provide a sensitivity analysis of the final profile to the alpha selection rule (e.g., by comparing L-curve-selected alpha with other criteria such as the discrepancy principle).
  4. [Sec. IV, Fig. 12 and text] The agreement between the Tikhonov reconstruction and the lambda-extrapolation method is presented as confirmation of reliability and even of 'well-posedness under optimal conditions.' However, both methods use the same lattice data and share overlapping authorship/collaboration (LPC); the comparison is a consistency check rather than an independent validation. The text should be moderated and should state this limitation explicitly.
minor comments (6)
  1. [Abstract] The phrase 'The reconstructed solutions is consistent' should read 'The reconstructed solutions are consistent.'
  2. [Eqs. (7)-(8)] Please define X_nx and Lambda_nlambda explicitly as vectors and clarify the matrix construction; the current notation using sets and an outer product is ambiguous.
  3. [Eq. (18)] The approximation (sigma_1 + alpha)/(sigma_n + alpha) ~ sigma_1/alpha requires sigma_n << alpha << sigma_1; please state this condition or revise the heuristic.
  4. [Sec. III and Sec. V] There are several typos: 'maskes' should be 'masks', 'achiev' should be 'achieves', and 'the uncertainty in lattice data do not grow linearly' should read 'the uncertainties in lattice data do not grow linearly.'
  5. [Appendix A] The proof is correct but uses nonstandard notation (C[x_min,x_max]); consider referring to C([x_min,x_max]) and citing a standard result for the density of polynomials in L2.
  6. [Sec. I and Sec. II.B] The statement that the inverse problem satisfies 'existence' is not fully established; the Paley-Wiener argument applies to the true solution, not to arbitrary data. Please clarify the meaning of existence in the Hadamard sense.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the Tikhonov reconstruction is benchmarked on a known toy-model ground truth and the uniqueness proof is in-paper; the only concern is a minor non-load-bearing self-citation overlap in the real-data cross-check.

full rationale

The derivation chain is self-contained. Ill-posedness is established directly from the SVD spectrum (Eq. 14, kappa~10^17) and Picard-criterion divergence (Eq. 15); Tikhonov regularization is a standard external method whose convergence bound (Eqs. 19-21) is cited from the textbook literature, not from the authors' prior work. The toy-model validation uses a synthetic ground truth ft(x)=6x(1-x), generates g(lambda) via the DFT, adds controlled noise, and reconstructs; this is an external benchmark, not a fitted input renamed as a prediction. The uniqueness theorem (Appendix A) is proved in the paper, so no uniqueness result is imported from the authors' previous papers. The real-data comparison with the lambda-extrapolation method (Ref. [163]) is a cross-check rather than an input: the Tikhonov solution is not constructed from the lambda-extrapolation output. That said, Ref. [163] shares author Jun Hua and the input data come from the LPC collaboration (Ref. [112]); this reduces the independent weight of the real-data agreement, so a minor non-load-bearing self-citation concern applies (score 2). The Sec. II.B assertion that the singular values 'decay more slowly than exponentially' is unsupported and arguably inconsistent with sigma1/sigma_n~10^17, but an unproved premise is a correctness risk, not a circular reduction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard inverse-problem mathematics plus several physics modeling assumptions: compact support and exponential decay of hadronic correlators, faithfulness of the discrete DFT discretization, and validity of the L-curve heuristic. The method introduces no new entities such as particles, forces, or conserved quantities. The free parameters are the regularization parameter, toy-model noise parameters, and the demonstration grid; none is fitted to an external physics target.

free parameters (4)
  • regularization parameter alpha = 7.31e-6 (pion quasi DA data)
    Selected by the L-curve criterion to balance residual and solution norms; it is data-dependent and directly controls the reconstructed distribution, so it is a free hyperparameter of the central demonstration.
  • toy noise amplitude sigma = 0.05
    Hand-chosen Gaussian noise level in the synthetic data (Sec. III) used to emulate lattice statistical errors; the demonstration's success depends on this choice.
  • toy signal-decay exponent delta_M = 0.02
    Hand-chosen exponential growth factor e^{delta_M |lambda|} modeling signal-to-noise degradation in lattice correlators (Sec. III, Eq. 24).
  • discretization grid (x_min, x_max, delta_x, lambda_max, delta_lambda) = (-2, 3, 0.01, 20, 0.5)
    The grid in Eq. (12) defines the illustrative condition number kappa ~ 1e17; the claimed instability and moderate-ill-posedness classification depend on this specific grid.
assumptions (5)
  • standard math Paley-Wiener: compactly supported L2 functions have entire Fourier transforms of exponential type
    Invoked in Sec. II.B to argue existence and analyticity of g(lambda) for partonic distributions with compact support.
  • domain assumption Partonic distributions have compact support in x and coordinate-space correlators decay exponentially as g(lambda) proportional to e^{-m_eff |z|}
    Assumed in Sec. II.B with references [78,97,98,176] to guarantee existence and uniqueness and to motivate the noise model; if violated, the solution space and convergence properties change.
  • domain assumption The discrete DFT matrix on the chosen grid faithfully represents the continuous limited Fourier inversion
    The condition number and SVD analysis (Sec. II.B) are performed on the discretized matrix K_re; the paper asserts discretization does not induce ill-posedness in the noise-free case (Sec. III), but the continuous-to-discrete correspondence is not rigorously established.
  • standard math Source condition f_true = K^dagger v with ||v||_l2 <= E, needed for the Tikhonov convergence bound
    Stated in Sec. II.C as a prerequisite for the convergence bound in Eqs. (19)-(21); not verified for the quasi-DA reconstructions.
  • domain assumption L-curve curvature maximum provides a near-optimal regularization parameter
    Adopted from the inverse-problem literature (Refs. [193,194]) and used throughout to fix alpha; standard but heuristic.

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Cite this review

Pith. "Pith review of Ill-Posedness in Limited Discrete Fourier Inversion and Regularization for Quasi Distributions in LaMET." pith.science (2026). https://pith.science/paper/XIH7I6JY

@misc{pith2026250616689,
  author       = {Pith},
  title        = {Pith review of: Ill-Posedness in Limited Discrete Fourier Inversion and Regularization for Quasi Distributions in LaMET},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XIH7I6JY}},
  note         = {Machine review of arXiv:2506.16689}
}
abstract

We systematically investigated the limited inverse discrete Fourier transform of the quasi distributions from the perspective of inverse problem theory. This transformation satisfies two of Hadamard's well-posedness criteria, existence and uniqueness of solutions, but critically violates the stability requirement, exhibiting exponential sensitivity to input perturbations. To address this instability, we implemented Tikhonov regularization with L-curve optimized parameters, demonstrating its validity for controlled toy model studies and real lattice QCD results of quasi distribution amplitudes. The reconstructed solutions is consistent with the physics-driven $\lambda$-extrapolation method. Our analysis demonstrates that the inverse Fourier problem within the large-momentum effective theory (LaMET) framework belongs to a class of moderately tractable ill-posed problems, characterized by distinct spectral properties that differ from those of more severely unstable inverse problems encountered in other lattice QCD applications. Tikhonov regularization establishes a rigorous mathematical framework for addressing the underlying instability, enabling first-principles uncertainty quantification without relying on ansatz-based assumptions.

Figures

Figures reproduced from arXiv: 2506.16689 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of the matrix [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Examples of discretization for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Dependence of the resulting solutions [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison with results from traditional SVD solu [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Numerical results for the pion quasi DA from the [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison with results from traditional SVD solu [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Dependence of regularized solutions on the regularization parameters [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Comparison with results of pion quasi DA from [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. Pion quasi DA from [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]

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Cited by 2 Pith papers

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    Using light-cone sum-rule input and Tikhonov-regularized inversion, the paper reconstructs the quark confining force in the proton and confirms an attractive, approximately linear-potential force at intermediate distances.

  2. Comment on "LaMET's Asymptotic Extrapolation vs. Inverse Problem"

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    The paper defends the inverse-problem view of LaMET reconstructions and argues that rigid parametric extrapolations underestimate PDF uncertainties when lattice data are noisy.

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Pith tools

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